In curve theory there is a long history of taking some interesting disconnection property and then studying the class of spaces determined by this property. In this thesis we study the spaces in which every countably infinite set disconnects. The disconnection number, Ds(X), of a connected space X is defined to be the smallest cardinal number &#954; such that X becomes disconnected upon removal of any set A with |A| = &#954; and |X \ A| &#8805; 2 provided such &#954; exists. We write X in D&#8501;0 if Ds(X) &#8804; &#8501;0 and call X a D&#8501;0-space. We write X in Dsw if X in D&#8501;0 and if each separator F of X between any two points a and b of X contains a separator between a and b consisting of finitely many points and call X a Dsw-space. Stone [St] obtained a characterization of connected, locally connected, separable, metric D&#8501;0-spaces. It is a corollary of Stone's theorem that every locally connected, separable, metric D&#8501;0-space X is a Dn-space for some integer n. Stone asked for an independent proof of this fact (i.e., one which does not rely on Stone's characterization theorem). We present a characterization theorem of these spaces and in the process we obtain an answer to Stone's question. We obtain a structure theorem for the class of connected, Hausdorff spaces in Dsw: If X is a connected, Hausdorff space in Dsw, then there exists a weaker topology for X which makes X a locally connected, Tychonoff, Dsw-space. Under this weaker topology X is the union of a rim-finite generalized R-tree and a finite set. If X is a connected, semi-colocally connected, separable metric Dsw-space, then X is hereditarily locally connected and, hence, X is the union of a R-tree and a finite set. If X is a non-degenerate, countably compact, connected, separable, Hausdorff, Dsw-space, then there exists a weaker topology for X which makes X a metric graph. For the class of non-metric continua in D&#8501;0 we give a characterization theorem as follows: A Hausdorff continuum X is a D&#8501;0-space if and only if X is a generalized graph. This generalizes a theorem of Nadler in the metric case. The connectivity degree of a space is introduced and its relation with disconnection number is discussed.